The absence of eigenvalues is proved for the Schrödinger operator −Δ+V(x,y)-Delta + V(x,y) in the Euclidean space whose potential is periodic in some variables and decays in the remaining variables faster than power 11. A similar result for the Maxwell operator is established.
{"title":"The absence of eigenvalues for certain operators with partially periodic coefficients","authors":"N. Filonov","doi":"10.1090/spmj/1730","DOIUrl":"https://doi.org/10.1090/spmj/1730","url":null,"abstract":"<p>The absence of eigenvalues is proved for the Schrödinger operator <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"negative normal upper Delta plus upper V left-parenthesis x comma y right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mo>−<!-- − --></mml:mo>\u0000 <mml:mi mathvariant=\"normal\">Δ<!-- Δ --></mml:mi>\u0000 <mml:mo>+</mml:mo>\u0000 <mml:mi>V</mml:mi>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>x</mml:mi>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mi>y</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">-Delta + V(x,y)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> in the Euclidean space whose potential is periodic in some variables and decays in the remaining variables faster than power <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"1\">\u0000 <mml:semantics>\u0000 <mml:mn>1</mml:mn>\u0000 <mml:annotation encoding=\"application/x-tex\">1</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. A similar result for the Maxwell operator is established.</p>","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48564417","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The sets of radial or nontangential limit points towards i∞iinfty of a Nevanlinna function qq are studied. Given a nonempty, closed, and connected subset L{mathcal {L}} of C+¯overline {{mathbb {C}}_+}, a Hamiltonian HH is constructed explicitly such that the radial and outer angular cluster sets towards i∞iinfty of the Weyl coefficient qHq_H
A mathematical model of the simplest three-link oscillatory gene network, the so-called repressilator, is considered. This model is a nonlinear singularly perturbed system of three ordinary differential equations. The existence and stability of a relaxation periodic solution invariant with respect to cyclic permutations of coordinates are investigated for this system.
{"title":"On a mathematical model of a repressilator","authors":"S. Glyzin, A. Kolesov, N. Rozov","doi":"10.1090/spmj/1727","DOIUrl":"https://doi.org/10.1090/spmj/1727","url":null,"abstract":"A mathematical model of the simplest three-link oscillatory gene network, the so-called repressilator, is considered. This model is a nonlinear singularly perturbed system of three ordinary differential equations. The existence and stability of a relaxation periodic solution invariant with respect to cyclic permutations of coordinates are investigated for this system.","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47451963","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A Banach limit on the space of all bounded real sequences is a positive normalized linear functional that is invariant with respect to the shift. The paper studies such properties of Banach limits as multiplicativity and the validity of Fubini’s theorem. A subset of Banach limits invariant with respect to dilation operators is also treated.
{"title":"Banach limits: extreme properties, invariance and the Fubini theorem","authors":"N. Avdeev, E. Semenov, A. Usachev","doi":"10.1090/spmj/1717","DOIUrl":"https://doi.org/10.1090/spmj/1717","url":null,"abstract":"A Banach limit on the space of all bounded real sequences is a positive normalized linear functional that is invariant with respect to the shift. The paper studies such properties of Banach limits as multiplicativity and the validity of Fubini’s theorem. A subset of Banach limits invariant with respect to dilation operators is also treated.","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41693367","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A possible way of proving the Riemann hypothesis consists of constructing a selfadjoint operartor whose spectrum coincides with the set {z:|Imz|>12,ζ(12−iz)=0}{z,: , |operatorname {Im}z|>frac 12, zeta big (frac {1}{2}-izbig )=0}. In the paper we construct a rank-one perturbation of a selfadjoint operator related to a certain canonical system for which a similar property is fulfilled.
{"title":"The set of zeros of the Riemann zeta function as the point spectrum of an operator","authors":"V. Kapustin","doi":"10.1090/spmj/1720","DOIUrl":"https://doi.org/10.1090/spmj/1720","url":null,"abstract":"<p>A possible way of proving the Riemann hypothesis consists of constructing a selfadjoint operartor whose spectrum coincides with the set <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-brace z colon StartAbsoluteValue upper I m z EndAbsoluteValue greater-than one half comma zeta left-parenthesis one half minus i z right-parenthesis equals 0 right-brace\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo>\u0000 <mml:mi>z</mml:mi>\u0000 <mml:mspace width=\"thinmathspace\" />\u0000 <mml:mo>:</mml:mo>\u0000 <mml:mspace width=\"thinmathspace\" />\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mo stretchy=\"false\">|</mml:mo>\u0000 </mml:mrow>\u0000 <mml:mi>Im</mml:mi>\u0000 <mml:mo><!-- --></mml:mo>\u0000 <mml:mi>z</mml:mi>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mo stretchy=\"false\">|</mml:mo>\u0000 </mml:mrow>\u0000 <mml:mo>></mml:mo>\u0000 <mml:mfrac>\u0000 <mml:mn>1</mml:mn>\u0000 <mml:mn>2</mml:mn>\u0000 </mml:mfrac>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mtext> </mml:mtext>\u0000 <mml:mi>ζ<!-- ζ --></mml:mi>\u0000 <mml:mstyle scriptlevel=\"0\">\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mo maxsize=\"1.2em\" minsize=\"1.2em\">(</mml:mo>\u0000 </mml:mrow>\u0000 </mml:mstyle>\u0000 <mml:mfrac>\u0000 <mml:mn>1</mml:mn>\u0000 <mml:mn>2</mml:mn>\u0000 </mml:mfrac>\u0000 <mml:mo>−<!-- − --></mml:mo>\u0000 <mml:mi>i</mml:mi>\u0000 <mml:mi>z</mml:mi>\u0000 <mml:mstyle scriptlevel=\"0\">\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mo maxsize=\"1.2em\" minsize=\"1.2em\">)</mml:mo>\u0000 </mml:mrow>\u0000 </mml:mstyle>\u0000 <mml:mo>=</mml:mo>\u0000 <mml:mn>0</mml:mn>\u0000 <mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">{z,: , |operatorname {Im}z|>frac 12, zeta big (frac {1}{2}-izbig )=0}</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. In the paper we construct a rank-one perturbation of a selfadjoint operator related to a certain canonical system for which a similar property is fulfilled.</p>","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48485301","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The oscillation equation for a singular string with discrete weight generated by a self-similar n n -link multiplier in the Sobolev space with a negative smoothness index is considered. It is shown that in the case of a noncompact multiplier, the string problem is equivalent to the spectral problem for an ( n − 1 ) (n-1) -periodic Jacobi matrix. In the case of n = 3 n=3 , a complete description of the spectrum of the problem is given, and a criterion for emergence of an eigenvalue in a gap of the continuous spectrum is obtained.
{"title":"Weighted string equation where the weight is a noncompact multiplier: continuous spectrum and eigenvalues","authors":"E. B. Sharov, I. Sheipak","doi":"10.1090/spmj/1723","DOIUrl":"https://doi.org/10.1090/spmj/1723","url":null,"abstract":"The oscillation equation for a singular string with discrete weight generated by a self-similar \u0000\u0000 \u0000 n\u0000 n\u0000 \u0000\u0000-link multiplier in the Sobolev space with a negative smoothness index is considered. It is shown that in the case of a noncompact multiplier, the string problem is equivalent to the spectral problem for an \u0000\u0000 \u0000 \u0000 (\u0000 n\u0000 −\u0000 1\u0000 )\u0000 \u0000 (n-1)\u0000 \u0000\u0000-periodic Jacobi matrix. In the case of \u0000\u0000 \u0000 \u0000 n\u0000 =\u0000 3\u0000 \u0000 n=3\u0000 \u0000\u0000, a complete description of the spectrum of the problem is given, and a criterion for emergence of an eigenvalue in a gap of the continuous spectrum is obtained.","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47808451","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The algebra H∞(D)H^infty (D) of bounded holomorphic functions on D⊂CDsubset mathbb {C} is projective free for a wide class of infinitely connected domains. In particular, for such DD every rectangular left-invertible matrix with entries in H∞(D)H^infty (D) can be extended in this class of matrices to an invertible square matrix. This follows from a new result on the structure of the maximal ideal space of H∞(D)H^infty (D) asserting that its covering dimension is
{"title":"Projective free algebras of bounded holomorphic functions on infinitely connected domains","authors":"A. Brudnyi","doi":"10.1090/spmj/1718","DOIUrl":"https://doi.org/10.1090/spmj/1718","url":null,"abstract":"<p>The algebra <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper H Superscript normal infinity Baseline left-parenthesis upper D right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:msup>\u0000 <mml:mi>H</mml:mi>\u0000 <mml:mi mathvariant=\"normal\">∞<!-- ∞ --></mml:mi>\u0000 </mml:msup>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>D</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">H^infty (D)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> of bounded holomorphic functions on <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper D subset-of double-struck upper C\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>D</mml:mi>\u0000 <mml:mo>⊂<!-- ⊂ --></mml:mo>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">C</mml:mi>\u0000 </mml:mrow>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">Dsubset mathbb {C}</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is projective free for a wide class of infinitely connected domains. In particular, for such <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper D\">\u0000 <mml:semantics>\u0000 <mml:mi>D</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">D</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> every rectangular left-invertible matrix with entries in <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper H Superscript normal infinity Baseline left-parenthesis upper D right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:msup>\u0000 <mml:mi>H</mml:mi>\u0000 <mml:mi mathvariant=\"normal\">∞<!-- ∞ --></mml:mi>\u0000 </mml:msup>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>D</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">H^infty (D)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> can be extended in this class of matrices to an invertible square matrix. This follows from a new result on the structure of the maximal ideal space of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper H Superscript normal infinity Baseline left-parenthesis upper D right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:msup>\u0000 <mml:mi>H</mml:mi>\u0000 <mml:mi mathvariant=\"normal\">∞<!-- ∞ --></mml:mi>\u0000 </mml:msup>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>D</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">H^infty (D)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> asserting that its covering dimension is <inl","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46137749","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Four special subsets of the Schwartz algebra are defined (this algebra consists of all entire functions of exponential type and of polynomial growth on the real axis). Perturbations of the zero sets for functions belonging to each of these subsets are studied. It is shown that the boundedness of the real part of the perturbing sequence is a sufficient and, generally speaking, unimprovable condition for preservation the subset from which the function in question is taken. An application of these results to spectral synthesis problems for differentiation-invariant subspaces of the Schwartz class on an interval of the real line is considered.
{"title":"Preservation of classes of entire functions defined in terms of growth restrictions along the real axis under perturbations of their zero sets","authors":"N. Abuzyarova","doi":"10.1090/spmj/1716","DOIUrl":"https://doi.org/10.1090/spmj/1716","url":null,"abstract":"Four special subsets of the Schwartz algebra are defined (this algebra consists of all entire functions of exponential type and of polynomial growth on the real axis). Perturbations of the zero sets for functions belonging to each of these subsets are studied. It is shown that the boundedness of the real part of the perturbing sequence is a sufficient and, generally speaking, unimprovable condition for preservation the subset from which the function in question is taken. An application of these results to spectral synthesis problems for differentiation-invariant subspaces of the Schwartz class on an interval of the real line is considered.","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44768528","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A Banach space XX is said to satisfy the positivity principle for small balls if for every finite Borel measures μmu and νnu on XX, the inequalities μ(B)≤ν(B)mu (B) leq nu (B) for all balls B of radius less than 1 imply that μ≤νmu leq nu. It is shown that no uniformly convex infinite-dimensional separable Banach space XX obeys the positivity principle for small balls.
A formula is given that makes it possible to reduce the calculation of an interpolation functor on a pair of local Morrey spaces to the calculation of this functor on pairs of vector function spaces constructed from the ideal spaces involved in the definition of the Morrey spaces in question. It is shown that a pair of local Morrey spaces is K K -monotone if and only if the pair of vector function spaces mentioned above is K K -monotone. This reduction makes it possible to obtain new interpolation theorems even for classical local spaces.
{"title":"On interpolation and 𝐾-monotonicity for discrete local Morrey spaces","authors":"E. Berezhnoi","doi":"10.1090/spmj/1707","DOIUrl":"https://doi.org/10.1090/spmj/1707","url":null,"abstract":"A formula is given that makes it possible to reduce the calculation of an interpolation functor on a pair of local Morrey spaces to the calculation of this functor on pairs of vector function spaces constructed from the ideal spaces involved in the definition of the Morrey spaces in question. It is shown that a pair of local Morrey spaces is \u0000\u0000 \u0000 K\u0000 K\u0000 \u0000\u0000-monotone if and only if the pair of vector function spaces mentioned above is \u0000\u0000 \u0000 K\u0000 K\u0000 \u0000\u0000-monotone. This reduction makes it possible to obtain new interpolation theorems even for classical local spaces.","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45203162","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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